We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the un...We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the unit speed. The models, without or with immigration, are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.展开更多
Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) ...Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.展开更多
We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in ...We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.展开更多
In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) su...In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.展开更多
Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual ...Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.展开更多
A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equati...A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.展开更多
In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for th...In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.展开更多
This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properti...This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.展开更多
It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued br...It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.展开更多
Let(Z_(n))_(n)≥0be a supercritical branching process in an independent and identically distributed random environment.We establish an optimal convergence rate in the Wasserstein-1 distance for the process(Z_(n))_(n)...Let(Z_(n))_(n)≥0be a supercritical branching process in an independent and identically distributed random environment.We establish an optimal convergence rate in the Wasserstein-1 distance for the process(Z_(n))_(n)≥0,which completes a result of Grama et al.[Stochastic Process.Appl.,2017,127(4):1255–1281].Moreover,an exponential nonuniform Berry-Esseen bound is also given.At last,some applications of the main results to the confidence interval estimation for the criticality parameter and the population size Znare discussed.展开更多
The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the sp...The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the spectral radius of Jacobi matrix of its generating function.展开更多
Let {Z_(n), n ≥ 0} be a supercritical branching process in an independent and identically distributed random environment ξ =(ξ_n)_(n≥0). In this paper, we get some deviation inequalities for ln(Z_(n+n_(0))/Z_(n_(0...Let {Z_(n), n ≥ 0} be a supercritical branching process in an independent and identically distributed random environment ξ =(ξ_n)_(n≥0). In this paper, we get some deviation inequalities for ln(Z_(n+n_(0))/Z_(n_(0))). And some applications are given for constructing confidence intervals.展开更多
Consider a branching process{Z_(n)}n≥0with immigration in varying environments.For a∈{0,1,2,...},let C(a)={n≥0:Z_n=a}be the collection of times at which the population size of the process attains level a.We give a ...Consider a branching process{Z_(n)}n≥0with immigration in varying environments.For a∈{0,1,2,...},let C(a)={n≥0:Z_n=a}be the collection of times at which the population size of the process attains level a.We give a criterion to determine whether the set C(a)is finite or not.For the critical Galton-Watson process,based on a moment method,we show that|C(a)∩[1,n]|/log n→S in distribution,where S is an exponentially distributed random variable with P(S>t)=e^(-t),t>0.展开更多
In this paper,a critical Galton-Watson branching process {Z_(n)} is considered.Large deviation rates of SZ_(n):=Σ_(i=1)^_(Z_(n)) Xi are obtained,where {X_(i);i≥1} is a sequence of independent and identically distrib...In this paper,a critical Galton-Watson branching process {Z_(n)} is considered.Large deviation rates of SZ_(n):=Σ_(i=1)^_(Z_(n)) Xi are obtained,where {X_(i);i≥1} is a sequence of independent and identically distributed random variables and X_(1) is in the domain of attraction of an-stable law with α∈(0,2).One shall see that the convergence rate is determined by the tail index of X_(1) and the variance of Z_(1).Our results can be compared with those ones of the supercritical case.展开更多
Let{X_(n)}_(n≥0) be a p-type(p≥2)supercritical branching process with immigration and mean matrix M.Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right ei...Let{X_(n)}_(n≥0) be a p-type(p≥2)supercritical branching process with immigration and mean matrix M.Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right eigenvectors v and u.Let ρ>1 and Y_(n)=ρ^(-n)[u·X_(n)-ρ^(n+1)-1/ρ-1(u·λ)],where the vector λ denotes the mean immigration rate.In this paper,we will show that Yn is a martingale and converges to an r.v.Y as n→∞.We study the rates of convergence to 0 as n→∞of P_(i)(|l·X_(n+1)/1·X_(n)-l·(X_(n)M)/1·X_(n)|>ε),P_(i)(|l·X_(n)/1·X_(n)-l·υ/1·υ>ε),P(|Y_(n)-Y|>ε) for any ε>0,i=1,…,p,1=(1,…,1)and l∈R^(p),the p-dimensional Euclidean space.It is shown that under certain moment conditions,the first two decay geometrically,while conditionally on the event Y≥α(α>0)supergeometrically.The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.展开更多
Let{Z(t),t≥0}be a continuous-time Markov branching process with immigration.In this paper,we mainly research the moment for Z(t).We calculate the specific expression of the moment M_(1) t and then M_(i)(t),i>can b...Let{Z(t),t≥0}be a continuous-time Markov branching process with immigration.In this paper,we mainly research the moment for Z(t).We calculate the specific expression of the moment M_(1) t and then M_(i)(t),i>can be found subsequently.The moments of the branching process provide great help for further studying the asymptotic behavior of the branching process,so it is indispensable to calculate the moment.展开更多
We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in deta...We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.展开更多
By generalizing a criterion of Mufa Chen for Markov jump processes,we establish the necessary and sufficient conditions for the extinction,explosion and coming down from infinity of a continuous-state nonlinear Neveu...By generalizing a criterion of Mufa Chen for Markov jump processes,we establish the necessary and sufficient conditions for the extinction,explosion and coming down from infinity of a continuous-state nonlinear Neveu’s branching process.展开更多
The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the asserti...The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the assertion of Rhyzhov and Skorokhod(Theory Probab.Appl.,1970)on the uniqueness of the solutions to the equation,which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.展开更多
Let(Zn)be a supercritical branching process with immigration in an independent and identically distributed random environment.Under necessary moment conditions,we show the exact convergence rate in the central limit t...Let(Zn)be a supercritical branching process with immigration in an independent and identically distributed random environment.Under necessary moment conditions,we show the exact convergence rate in the central limit theorem on log Zn and establish the corresponding local limit theorem by using the moments of the natural submartingale and the convergence rates of its logarithm.By similar approach and with the help of a change of measure,we also present the so-called integrolocal theorem and integral large deviation theorem to characterize the precise asymptotics of the upper large deviations.展开更多
基金supported by the National Key R&D Program of China(2020YFA0712901).
文摘We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is determined at its birth and its remaining lifetime decreases at the unit speed. The models, without or with immigration, are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.
基金partially supported by the National Nature Science Foundation of China(11601286,11501146)。
文摘Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.
基金supported by the National Natural Science Foundation of China(11571052,11731012)the Hunan Provincial Natural Science Foundation of China(2018JJ2417)the Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering(2018MMAEZD02)。
文摘We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.
文摘In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.
文摘Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.
基金supported by the National Key R&D Program of China(2020YFA0712900)the National Natural Science Foundation of China(11531001).
文摘A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.
文摘In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.
基金supported by NNSF of China(6053408070571079)Open Fundation of SKLSE of Wuhan University (2008-07-03)
文摘This article deals with some properties of Galton-Watson branching processes in varying environments. A necessary and suffcient condition for relative recurrent state is presented, and a series of ratio limit properties of the transition probabilities are showed.
文摘It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.
基金Supported by the National Natural Science Foundation of China(Grant No.11971063)CY Initiative of Excellence(Grant No.“Investissements d’Avenir”ANR-16-IDEX-0008)Project“EcoDep”(Grant No.PSI-AAP2020-0000000013)。
文摘Let(Z_(n))_(n)≥0be a supercritical branching process in an independent and identically distributed random environment.We establish an optimal convergence rate in the Wasserstein-1 distance for the process(Z_(n))_(n)≥0,which completes a result of Grama et al.[Stochastic Process.Appl.,2017,127(4):1255–1281].Moreover,an exponential nonuniform Berry-Esseen bound is also given.At last,some applications of the main results to the confidence interval estimation for the criticality parameter and the population size Znare discussed.
文摘The spectral radiuses of Galton-Watson branching processes which describes the speed of the process escaping from any state are calculated. Under the condition of irreducibility,it is show that this is equal to the spectral radius of Jacobi matrix of its generating function.
文摘Let {Z_(n), n ≥ 0} be a supercritical branching process in an independent and identically distributed random environment ξ =(ξ_n)_(n≥0). In this paper, we get some deviation inequalities for ln(Z_(n+n_(0))/Z_(n_(0))). And some applications are given for constructing confidence intervals.
基金Supported by National Natural Science Foundation of China(Grant No.11501008)Nature Science Foundation of Anhui Educational Committee(Grant No.2023AH040025)。
文摘Consider a branching process{Z_(n)}n≥0with immigration in varying environments.For a∈{0,1,2,...},let C(a)={n≥0:Z_n=a}be the collection of times at which the population size of the process attains level a.We give a criterion to determine whether the set C(a)is finite or not.For the critical Galton-Watson process,based on a moment method,we show that|C(a)∩[1,n]|/log n→S in distribution,where S is an exponentially distributed random variable with P(S>t)=e^(-t),t>0.
基金supported by the National Natural Science Foundation of China(No.12101023,No.11871103 and 12271043)the National Key Research and Development Program of China(No.2020YFA0712900)Fundamental Research Funds for the Central Universities(No.2023MS077).
文摘In this paper,a critical Galton-Watson branching process {Z_(n)} is considered.Large deviation rates of SZ_(n):=Σ_(i=1)^_(Z_(n)) Xi are obtained,where {X_(i);i≥1} is a sequence of independent and identically distributed random variables and X_(1) is in the domain of attraction of an-stable law with α∈(0,2).One shall see that the convergence rate is determined by the tail index of X_(1) and the variance of Z_(1).Our results can be compared with those ones of the supercritical case.
基金Supported by the National Natural Sciences Foundations of China(Grants Nos.11771452,11571372)Sciences Foundations of Hunan(Grants No.2017JJ2328)。
文摘Let{X_(n)}_(n≥0) be a p-type(p≥2)supercritical branching process with immigration and mean matrix M.Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right eigenvectors v and u.Let ρ>1 and Y_(n)=ρ^(-n)[u·X_(n)-ρ^(n+1)-1/ρ-1(u·λ)],where the vector λ denotes the mean immigration rate.In this paper,we will show that Yn is a martingale and converges to an r.v.Y as n→∞.We study the rates of convergence to 0 as n→∞of P_(i)(|l·X_(n+1)/1·X_(n)-l·(X_(n)M)/1·X_(n)|>ε),P_(i)(|l·X_(n)/1·X_(n)-l·υ/1·υ>ε),P(|Y_(n)-Y|>ε) for any ε>0,i=1,…,p,1=(1,…,1)and l∈R^(p),the p-dimensional Euclidean space.It is shown that under certain moment conditions,the first two decay geometrically,while conditionally on the event Y≥α(α>0)supergeometrically.The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.
基金supported by the National Natural Science Foundation of China(No.11901392).
文摘Let{Z(t),t≥0}be a continuous-time Markov branching process with immigration.In this paper,we mainly research the moment for Z(t).We calculate the specific expression of the moment M_(1) t and then M_(i)(t),i>can be found subsequently.The moments of the branching process provide great help for further studying the asymptotic behavior of the branching process,so it is indispensable to calculate the moment.
基金supported by National Natural Science Foundations of China (Grant Nos. 10771216 and 11071259)
文摘We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.
基金Supported by NSFC(Grant No.12061004)NSF of Ningxia(Grant No.2021AAC02018)+2 种基金the Fundamental Research Funds for the Central Universities,North Minzu University(Grant No.2020KYQD17)Major research project for North Minzu University(Grant No.ZDZX201902)the Construction Project of First-Class Disciplines in Ningxia Higher Education(Grant No.NXYLXK2017B09)。
文摘By generalizing a criterion of Mufa Chen for Markov jump processes,we establish the necessary and sufficient conditions for the extinction,explosion and coming down from infinity of a continuous-state nonlinear Neveu’s branching process.
基金supported by the National Key R&D Program of China(Grant No.2020YFA0712900)the National Natural Science Foundation of China(Grant No.12271029)。
文摘The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the assertion of Rhyzhov and Skorokhod(Theory Probab.Appl.,1970)on the uniqueness of the solutions to the equation,which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.
基金Supported by Shandong Provincial Natural Science Foundation(Grant No.ZR2021MA085)National Natural Science Foundation of China(Grant No.11971063)。
文摘Let(Zn)be a supercritical branching process with immigration in an independent and identically distributed random environment.Under necessary moment conditions,we show the exact convergence rate in the central limit theorem on log Zn and establish the corresponding local limit theorem by using the moments of the natural submartingale and the convergence rates of its logarithm.By similar approach and with the help of a change of measure,we also present the so-called integrolocal theorem and integral large deviation theorem to characterize the precise asymptotics of the upper large deviations.
